[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-125195-en":3,"doc-seo-125195-105":30,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":4,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":27,"seo_description":14,"update_tm":28,"read_time":29},125195,1374391975076,"Riley","https://ap-avatar.wpscdn.com/avatar/14000253ca4ec9f6853?x-image-process=image/resize,m_fixed,w_180,h_180&k=1783305029341752051",8,"Research & Report","Recursive Optimization - Exact and Efficient Combinatorial Optimization Algorithm Design Principles with Applications to Machine Learning","This thesis presents a generic algorithm design framework for solving combinatorial optimization problems, subsuming classical recursive optimization methods such as greedy, dynamic programming, divide-and-conquer, and branch-and-bound (BnB). It focuses on machine learning applications and formalizes the branch-and-bound approach within a unified design methodology. Grounded in Bird’s algebra of programming via Haskell code, the framework integrates constructive algorithmics, combinatorial generation, and combinatorial geometry to yield sound, concise, efficient algorithms. The work analyzes four core ML problems—classification, clustering, decision trees, and empirical risk minimization for ReLU networks—showing polynomial-time solvability. Experiments demonstrate provably exact solutions with substantially lower computation time than state-of-the-art BnB methods on selected example problems.","RECURSIVE OPTIMIZATION: EXACT AND EFFICIENT COMBINATORIAL OPTIMIZATION ALGORITHM DESIGN PRINCIPLES WITH APPLICATIONS TO  \nMACHINE LEARNING  \nBy  \nXi He  \nA thesis submitted to the University of Birmingham for the degree of  \nDOCTOR OF PHILOSOPHY  \nSchool of Computer science  \nCollege of Engineering and Physical Sciences  \nUniversity of Birmingham  \nApril 24, 2025  \nUniversity of Birmingham Research Archive  \ne-theses repository  \nThis unpublished thesis/dissertation is copyright of the author and/or third parties. The intellectual property rights of the author or third parties in respect of this work are as defined by The Copyright Designs and Patents Act 1988 or as modified by any successor legislation.  \nAny use made of information contained in this thesis/dissertation must be in accordance with that legislation and must be properly acknowledged. Further distribution or reproduction in any format is prohibited without the permission of the copyright holder.  \nAbstract  \nThis thesis presents a generic algorithm design framework for solving combinatorial optimization problems, it subsumes the classical recursive optimization methods, such as greedy, dynamic programming, divide-and-conquer, and branch-and-bound (BnB) methods. In particular, this thesis focuses on solving combinatorial optimization problems in machine learning to demonstrate the effectiveness and practicality of this framework.  \nOur framework is grounded in Bird’s theory of the algebra of programming, which is a relational formalism for deriving correct-by-construction algorithms from specifications. We introduce this theory through Haskell code, with a particular emphasis on its application to combinatorial optimization. Additionally, wereformulate the branch-and-bound method to integrate it formally into this framework, thereby establishing a unified approach for designing recursive combinatorial optimization algorithms.  \nMore broadly, our theoretical foundation is an integration of constructive algorithmics (or transformational programming), combinatorial generation, and combinatorial geometry. These topics are integrated together to achieve our final goal—designing efficient combinatorial optimization algorithms, particularly for machine learning problems. They are interconnected in such a way that both geometric algorithms for addressing fundamental combinatorial geometry problems and efficient combinatorial generators can be structured or reformulated systematically using principles from constructive algorithmics. This approach facilitates the design of efficient (in terms of worse-case complexity and parallelizability) geometric algorithms and combinatorial generators that are sound and concise. Moreover, the geometric insights allow us to reveal the combinatorial properties of combinatorial problems, which allows us to significantly simplify the combinatorial complexity of the problem.  \nOur main contribution does not lie in advancing the three themes in the framework, although we have novel contributions to each of them. Instead, the primary contribution of this thesis lies in how to integrate these themes for solving combinatorial optimization problems in machine learning. To demonstrate the effectiveness of our framework, we address four fundamental problems in machine learning: classification, clustering, decision tree, and empirical risk minimization for ReLU neural networks. We provide a detailed analysis of the combinatorial properties of these problems, demonstrating that these problems can be solved in polynomial time. To the best of our knowledge, all algorithms we propose are the fastest known in terms of worst-case complexity, and their performance can be further improved through the acceleration techniques we introduce.  \nFinally, two example problems (the 0-1 loss linear classification problem and the K-medoids problem) are selected to show end-to-end implementations in Haskell. In our experiments, we compare our algorithm with state-of-the-","cbCaijkysj8Fk63v","https://ap.wps.com/l/cbCaijkysj8Fk63v","pdf",2210986,1,279,"English","en",105,"# Abstract\n## Framework and theoretical foundation\n## Machine learning applications and complexity results\n## Experimental evaluation and comparisons\n## Interpretable machine learning contributions","[{\"question\":\"What is the main goal of this thesis on recursive optimization?\",\"answer\":\"To develop a generic algorithm design framework for solving combinatorial optimization problems exactly and efficiently, with emphasis on machine learning applications.\"},{\"question\":\"How does the framework unify classical recursive methods and branch-and-bound?\",\"answer\":\"It is grounded in Bird’s algebra of programming and formally integrates branch-and-bound into the framework so recursive combinatorial optimization algorithms can be designed in a unified way.\"},{\"question\":\"Which machine learning problems are analyzed and what complexity claims are made?\",\"answer\":\"Classification, clustering, decision trees, and empirical risk minimization for ReLU neural networks are analyzed, and the thesis demonstrates these problems can be solved in polynomial time.\"}]","Recursive Optimization - Exact and Efficient Combinatorial Optimization Algorithm Design Principles with Applications to Machine Learning | PDF",1785897331,703,{"code":4,"msg":31,"data":32},"ok",{"site_id":24,"language":23,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"recursive-optimization-exact-and-efficient-combinatorial-optimization-algorithm-design-principles-with-applications-to-machine-learning","",{"@graph":36,"@context":85},[37,54,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":53},"https://docshare.wps.com/document/recursive-optimization-exact-and-efficient-combinatorial-optimization-algorithm-design-principles-with-applications-to-machine-learning/125195/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":23,"description":14,"dateModified":62,"datePublished":62,"encodingFormat":61,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":41,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-08-05",true,{"@type":65,"interactionType":66,"userInteractionCount":4},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What is the main goal of this thesis on recursive optimization?","Question",{"text":75,"@type":76},"To develop a generic algorithm design framework for solving combinatorial optimization problems exactly and efficiently, with emphasis on machine learning applications.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the framework unify classical recursive methods and branch-and-bound?",{"text":80,"@type":76},"It is grounded in Bird’s algebra of programming and formally integrates branch-and-bound into the framework so recursive combinatorial optimization algorithms can be designed in a unified way.",{"name":82,"@type":73,"acceptedAnswer":83},"Which machine learning problems are analyzed and what complexity claims are made?",{"text":84,"@type":76},"Classification, clustering, decision trees, and empirical risk minimization for ReLU neural networks are analyzed, and the thesis demonstrates these problems can be solved in polynomial time.","https://schema.org",{"og:url":52,"og:type":87,"og:title":13,"og:site_name":59,"og:description":14},"article",{"robots":89,"canonical":52},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":20,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":53,"doc_module":4,"doc_module_name":46,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":46,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":46,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":46,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]